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Parikh vectors
5TFD_1 1FNZ_1 2LLZ_1 Letter Amino acid
5 14 3 P Proline
2 6 0 W Tryptophan
8 6 7 Y Tyrosine
10 13 10 D Aspartic acid
4 0 1 C Cysteine
20 7 5 E Glutamic acid
18 9 4 K Lycine
11 18 8 F Phenylalanine
21 24 11 S Serine
13 20 4 T Threonine
3 1 1 H Histidine
12 21 5 V Valine
20 19 3 G Glycine
18 15 7 I Isoleucine
24 19 10 L Leucine
8 1 4 M Methionine
12 17 4 A Alanine
8 3 5 R Arginine
6 16 6 N Asparagine
6 8 2 Q Glutamine 

5TFD_1|Chain A|Cystic fibrosis transmembrane conductance regulator|Homo sapiens (9606)
>1FNZ_1|Chain A|BARK AGGLUTININ I, POLYPEPTIDE A|Robinia pseudoacacia (35938)
>2LLZ_1|Chain A|Uncharacterized protein yjdK|Escherichia coli (83333)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
5TFD , Knot 107 229 0.84 40 156 224 
SLTTTEVVMENVTAFWEEGGTPVLKDINFKIERGQLLAVAGSTGAGKTSLLMMIMGELEPSEGKIKHSGRISFCSQFSWIMPGTIKENIIFGVSYDEYRYRSVIKACQLEEDISKFAEKDNIVLGEGGITLSGGQRARISLARAVYKDADLYLLDSPFGYLDVLTEKEIFESCVCKLMANKTRILVTSKMEHLKKADKILILHEGSSYFYGTFSELQNLQPDFSSKLMG
1FNZ , Knot 107 237 0.82 38 152 227 
TGSLSFSFPKFAPNQPYLINQGDALVTSTGVLQLTNVVNGVPSSKSLGRALYAAPFQIWDSTTGNVASFVTSFTFIIQAPNPATTADGLAFFLAPVDTQPLDLGGMLGIFKDGYFNKSNQIVAVEFDTFSNGDWDPKGRHLGINVNSIESIKTVPWNWTNGEVANVFISYEASTKSLTASLVYPSLETSFIIDAIVDVKIVLPEWVRFGFSATTGIDKGYVQTNDVLSWSFESNLPG
2LLZ , Knot 52 100 0.79 38 87 98 
GHMEGKNKFNTYVVSFDYPSSYSSVFLRLRSLMYDMNFSSIVADEYGIPRQLNENSFAITTSLAASEIEDLIRLKCLDLPDIDFDLNIMTVDDYFRQFYK

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(5TFD_1)}(2) \setminus P_{f(1FNZ_1)}(2)|=82\), \(|P_{f(1FNZ_1)}(2) \setminus P_{f(5TFD_1)}(2)|=78\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:0100001110010111001101110010101001011111100111000111111101010010100010101000101111101000111110000000001101001000100110000111101110101100101011011000101011001110101100001100010011100001110001001001001111001000101010010010101000111
Pair \(Z_2\) Length of longest common subsequence
5TFD_1,1FNZ_1 160 4
5TFD_1,2LLZ_1 145 3
1FNZ_1,2LLZ_1 163 3

Newick tree

 
[
	1FNZ_1:83.32,
	[
		5TFD_1:72.5,2LLZ_1:72.5
	]:10.82
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{466 }{\log_{20} 466}-\frac{229}{\log_{20}229})=68.6\)
Status Protein1 Protein2 d d1/2
Query variables 5TFD_1 1FNZ_1 87 87
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]